59 
of Edinburgh, Session 1869-70. 
write the product in the column containing p 2 , q 2 . We then 
multiply the newly found A by the p above it; the preceding A 
by its q, that is in this case 3.2 and 1.1, and write the sum 7 as 
the third value of A. Again, taking the sum of the products 
jp 3 A 3 , q, 2 A 2 , and, as we may call it for generality’s sake, r 1 A 1 , we 
have 2.7 + 2.2 + 1.1 = 19 for A 4 . In this way we obtain the 
successive values of A. 
The values of B are found in the same way, observing that 
B x = 0, B a = 1. So also are the values of C, and if it be wished, 
those of D, E, F, &c., obtained, the first effective term being de- 
layed a step, as shown in the scheme. 
This method was applied to the three irrational quantities, log 5, 
log 3, and log 2 ; and the results were used in explaining the doc- 
trine of musical temperaments. 
When two quantities only are compared, it is well known that 
the cross products of the adjoining fractions differ by unit, or that, 
taking three contiguous terms, such as — 
^- 3 , ^ 5 , we have the equation, 
^3 B 4 Bg 
A 3 B 4 - A 4 B 3 = - A 4 B 5 + A 5 B 4 , 
which may be expressed, according to Cayley’s notation of deter- 
minants — 
| A 3 a., I 
I A 4 Ag 1 
1 B, B 4 1 - 
1 B, Bg | 
In the very same way, when three magnitudes are compared, 
we have the equation — 
A 3 A 4 Ag 
K Ag A, 
b 3 b 4 b 3 
== 4 - 
B 4 “ Bg Bg 
0 3 C 4 Cg 
C 4 Cg Cg 
that is to say, this determinant is unit throughout. 
The extension of this method to more than three quantities is 
easy. In conclusion, an opinion was expressed, that as the Brounc- 
kerian process applied to two magnitudes has already thrown great 
light on the doctrine of squares, this extension of it may be 
expected to do as much for the still higher departments of the 
theory of numbers. 
